In the present paper we study types and intersecting definable families of sets in connection with a notion of definable compactness that applies to definable topological spaces. A definable topological space in a structure \(\mathcal {M}\)  is a topological space \((X,\tau )\) , with \(X\subseteq M^n\) , such that there exists a basis for \(\tau \)  that is (uniformly) definable. The study of tame definable topological spaces is connected to the development of o-minimality and dates back to Pillay [1]}. A prominent example in o-minimal structures, beyond the canonical o-minimal euclidean topology, is given by definable groups [2]}, which admit a definable manifold structure in the sense of van den Dries [3]} making them topological groups. Further examples within o-minimality are given by definable families of functions with definable norms [4]}, definable metric spaces [5]}[6]}, and some classical non-metrizable spaces in topology [7]}. The first notion of definable compactness in o-minimal structures was introduced for definable manifold spaces, albeit the definition generalizes to any o-minimal definable topology, by Peterzil and Steinhorn [8]}, and proved to be equivalent in the euclidean topology to being closed and bounded. This is the property that every definable curve converges, which we refer to in this paper as curve-compactness. A more recent approach with wider applicability (including outside linearly ordered structures) to the notion, and the one we use in this paper, is as follows: every downward directed definable family of nonempty closed sets has nonempty intersection (a family is downward directed if for any two sets in the family there is a third one that is a subset of their intersection). This property has been studied in recent years in a general model theoretic context by Fornasiero [9]}. In the context of definable topologies in o-minimal structures, it has been approached by Johnson [10]}, and by Thomas, Walsberg and the author [11]}. Johnson [10]} noted that, within o-minimality, it is equivalent to curve-compactness for euclidean sets. In [11]}, we proved this equivalence for all definable topologies in o-minimal expansions of ordered groups. Additionally we proved, in the case of an o-minimal expansion of an ordered field, that definable compactness is equivalent to type-compactness: for every definable type that inhabits the topological space there is a point contained in every closed set in the type. From this we derived the equivalence with classical topological compactness in o-minimal expansions of the field of reals.
